## Tag `02AW`

Chapter 102: Exercises > Section 102.49: Schemes, Final Exam, Spring 2009

Exercise 102.49.1. Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point. Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.

- Show that $x$ is a closed point of $X$.
- Show that $H^0(X, \mathcal{F})$ is not zero.
- Show that $\mathcal{F}$ is generated by global sections.
- Show that $H^p(X, \mathcal{F}) = 0$ for $p > 0$.

The code snippet corresponding to this tag is a part of the file `exercises.tex` and is located in lines 4925–4937 (see updates for more information).

```
\begin{exercise}
\label{exercise-Noetherian-coherent}
Let $X$ be a Noetherian scheme.
Let $\mathcal{F}$ be a coherent sheaf on $X$.
Let $x \in X$ be a point.
Assume that $\text{Supp}(\mathcal{F}) = \{ x \}$.
\begin{enumerate}
\item Show that $x$ is a closed point of $X$.
\item Show that $H^0(X, \mathcal{F})$ is not zero.
\item Show that $\mathcal{F}$ is generated by global sections.
\item Show that $H^p(X, \mathcal{F}) = 0$ for $p > 0$.
\end{enumerate}
\end{exercise}
```

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